r. j. poole and m. p. escudier dept. engineering, mechanical engineering, university of liverpool
DESCRIPTION
3D FLOW OF VISCOELASTIC FLUIDS OVER A BACKWARD-FACING STEP PRECEDED BY A GRADUAL CONTRACTION. A. Afonso Centro de Estudos de Fenómenos de Transporte, DEMEGI Faculdade de Engenharia, Universidade do Porto, Portugal, [email protected]. F. T. Pinho - PowerPoint PPT PresentationTRANSCRIPT
R. J. Poole and M. P. EscudierDept. Engineering, Mechanical Engineering, University of LiverpoolLiverpool L69 3GH, UK, [email protected],[email protected]
3D FLOW OF VISCOELASTIC FLUIDS OVER A 3D FLOW OF VISCOELASTIC FLUIDS OVER A
BACKWARD-FACING STEP PRECEDED BY A BACKWARD-FACING STEP PRECEDED BY A
GRADUAL CONTRACTIONGRADUAL CONTRACTION
A. AfonsoCentro de Estudos de Fenómenos de Transporte, DEMEGI Faculdade de Engenharia, Universidade do Porto, Portugal, [email protected]
F. T. PinhoCentro de Estudos de Fenómenos de Transporte, Dep. Eng. MecânicaEscola de Engenharia, Universidade do Minho, Portugal, [email protected]
AERC 200522nd to 24th April 2005 Grenoble, France
3D Viscoelastic flow in smooth contractions European Congress on Rheology
AERC 2005
Flow geometry
Experiments of Poole et al (2004) with solutions of PAA
Area ratio R = d/D = 0.7
Upstream spanwise velocity profiles (x-z plane) at x/h=-8.33 and 0
d = 28mm, h = 6mm,
D = 40mm, w = 80mm
Inlet duct: 120 DH long
(area ratio > 2/3 double backward-facing step )
Aspect ratios A1 = w/h = 13.3
A2 = w/d= 2.86
3D Viscoelastic flow in smooth contractions European Congress on Rheology
AERC 2005
Experimental and numerical findings
Figure : Spanwise variation of mean streamwise velocity ( U/UB ) profiles for 0.1% PAA
z / w
U/UB
0 0.25 0.5 0.75 10
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
NewtonianCarreau-YasudaPTT
x / h = 0
0.1% PAA
x / h = -8.33
0.1% PAA Re 120
Cat’s ears
Spanwise variation at y/D=0.5GNF
PTT(N2=0)
3D Viscoelastic flow in smooth contractions European Congress on Rheology
AERC 2005
Experimental and numerical findings 3
0.1% PAA Re 120Downstream
Figure : Mean streamwise velocity ( U/UB ) profiles for 0.1% PAA
y/D
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
PTTx/h0 1 2 3 4
1
U = 0
XR = 2.3
Newtonian Carreau-Yasuda
3D Viscoelastic flow in smooth contractions European Congress on Rheology
AERC 2005
Objective
Cat’s ears: Why?
Shear-thinning:No
Elasticity - : No Ψ1
Qualitative calculation with PTT: parametric investigation
Effect of
Effect of De
Effect of
Effect of Re
Ψ2
ηe
Individual and combined effects
3D Viscoelastic flow in smooth contractions European Congress on Rheology
AERC 2005
Governing equations
∂∂tJρui( ) +
∂
∂ξ lρβ lj %u jui( ) = −β li
∂p
∂ξ l+
∂
∂ξ lβ lj %τ ij , p + β ljτ ij ,s( ) + +Jρgi
1) Mass
2) Momentum
3) Constitutive equation
€
1+λε
η pτ kk,p
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟τ ij,p + λ
∂τ ij,p∂t
+∂ ukτ ij,p( )
∂xk
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟=η p
∂ui∂x j
+∂u j∂xi
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟+ λ τ jk,p
∂ui∂xk
+ τ ik,p∂u j∂xk
⎛
⎝ ⎜
⎞
⎠ ⎟−
€
λξ2τ jk,p
∂ui∂xk
+∂uk∂xi
⎛
⎝ ⎜
⎞
⎠ ⎟+ τ ik,p
∂u j∂xk
+∂uk∂x j
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
Full PTT (linear stress coefficient)
τ ij = 2η sDij + τ ij , p
Newtonian solvent
3D Viscoelastic flow in smooth contractions European Congress on Rheology
AERC 2005
Numerical method: brief description
2) Structured, colocated and non-orthogonal meshes
3) Momentum (ui)
∂∂t
Jρui( )+∂
∂ξlρβlj ˜ u j ˆ u i( )−
∂∂ξl
ηp
Jβljβlj
∂ui∂ξl
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟ −
∂ βljτij,s( )
∂ξl=−βli
∂p∂ξl
+∂
∂ξlβlj˜ τ ij( )+Jρgi −
∂∂ξl
ηp
Jβljβlj
∂ui∂ξl
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
solvent polymer
1) Finite volume method (Oliveira et al,1998; Oliveira & Pinho, 1999)
4) Discretization (formally 2nd order)Diffusive terms: central differences (CDS)Advective terms: CUBISTA (deferred correction)
(Alves et al, 2000, 2003)
5) Special formulations for cell-face velocities and stresses
3D Viscoelastic flow in smooth contractions European Congress on Rheology
AERC 2005
Computational domain and mesh
5 m (62 DH)
120 h
20 cells
30 cells
102 000 total cells1 020 000 DF
3D Viscoelastic flow in smooth contractions European Congress on Rheology
AERC 2005
Inlet flow
x/h=-16
ξ =0.001 ξ =0.2
ε =0.05,λ = 0.1 s,η p = 0.5 Pa.s,η s = 0.05 Pa.s
3D Viscoelastic flow in smooth contractions European Congress on Rheology
AERC 2005
Non-dimensional numbers
Re'= ρUbhηs+ηp( )
Reynolds number
De=Ubλp
hDeborah number
Re=ρUbhμc
&γC =Ubh
with and
ε
ξ
Extensional parameter
Slip parameter
Bulk velocity at contraction exit
3D Viscoelastic flow in smooth contractions European Congress on Rheology
AERC 2005
Effect of ξ: 1
Re '=5.14
kitten’s ears
Absence of kitten’s ears
Several values of ε
ε =0.15
3D Viscoelastic flow in smooth contractions European Congress on Rheology
AERC 2005
Effect of ξ: 2
Re '=5.14
kitten’s ears:high De, high ξ, low ε
3D Viscoelastic flow in smooth contractions European Congress on Rheology
AERC 2005
Effect of ξ: 3
Re '=5.14x h=−8.0;−4.0;−2.06;−0.1
kitten’s ears
x/h=-8
x/h=-4
x/h=-2.06
x/h=-0.1
Effect ofinertia
3D Viscoelastic flow in smooth contractions European Congress on Rheology
AERC 2005
Effect of εRe '=5.14 ξ =0.2
Closed symbols: kitten’s ears
x/h=-8
x/h=-4
x/h=-2.06
x/h=-0.1
(a) (b)
Effect of De (next slide)
3D Viscoelastic flow in smooth contractions European Congress on Rheology
AERC 2005
Effect of De
Re '=5.14 ξ =0.2
ε =0.025ε =0.1
DeDe
3D Viscoelastic flow in smooth contractions European Congress on Rheology
AERC 2005
Effect of Re’: 1
ξ =0.2 ε =0.005De=19.6 De=39.2
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Re=0.6 Re’=0.48 Re=0.6 Re’=0.43
3D Viscoelastic flow in smooth contractions European Congress on Rheology
AERC 2005
Effect of Re’: 2
ξ =0.2 ε =0.005De=19.6 De=39.2
Re=1.7 Re’=1.4 Re=1.7 Re’=1.3
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3D Viscoelastic flow in smooth contractions European Congress on Rheology
AERC 2005
Effect of Re’: 3
ξ =0.2 ε =0.005De=19.6 De=39.2
Re=3.4 Re’=2.8 Re=3.4 Re’=2.6
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QuickTime™ and aBMP decompressor
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3D Viscoelastic flow in smooth contractions European Congress on Rheology
AERC 2005
Effect of Re’: 4
ξ =0.2 ε =0.005De=19.6 De=39.2
Re=6.3 Re’=5.2 Re=6.3 Re’=4.7
QuickTime™ and aBMP decompressor
are needed to see this picture.
QuickTime™ and aBMP decompressor
are needed to see this picture.
3D Viscoelastic flow in smooth contractions European Congress on Rheology
AERC 2005
Conclusions
•Cat’s ears are qualitatively predicted by PTT (kitten’s ears)
•N2≠ 0 (essential)— high ξ
•Low ε
•High De
•Intermediate Re
•Sometimes enhanced peaks observed at corners
•Low Re: very slim profiles at contraction exit, no peaks
•High Re: flat profiles at contracton exit, no peaks
•Accurate predictions: different transient properties ???